Average Error: 0.1 → 0.0
Time: 46.1s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5.0\]
↓
\[x \cdot t + \mathsf{fma}\left(5.0, y, \left(y + z\right) \cdot x + \left(y + z\right) \cdot x\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5.0
↓
x \cdot t + \mathsf{fma}\left(5.0, y, \left(y + z\right) \cdot x + \left(y + z\right) \cdot x\right)
double f(double x, double y, double z, double t) {
        double r6509126 = x;
        double r6509127 = y;
        double r6509128 = z;
        double r6509129 = r6509127 + r6509128;
        double r6509130 = r6509129 + r6509128;
        double r6509131 = r6509130 + r6509127;
        double r6509132 = t;
        double r6509133 = r6509131 + r6509132;
        double r6509134 = r6509126 * r6509133;
        double r6509135 = 5.0;
        double r6509136 = r6509127 * r6509135;
        double r6509137 = r6509134 + r6509136;
        return r6509137;
}
↓
double f(double x, double y, double z, double t) {
        double r6509138 = x;
        double r6509139 = t;
        double r6509140 = r6509138 * r6509139;
        double r6509141 = 5.0;
        double r6509142 = y;
        double r6509143 = z;
        double r6509144 = r6509142 + r6509143;
        double r6509145 = r6509144 * r6509138;
        double r6509146 = r6509145 + r6509145;
        double r6509147 = fma(r6509141, r6509142, r6509146);
        double r6509148 = r6509140 + r6509147;
        return r6509148;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5.0\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{\left(x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + x \cdot t\right)} + y \cdot 5.0\]

  4. Applied associate-+l+0.1

    \[\leadsto \color{blue}{x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + \left(x \cdot t + y \cdot 5.0\right)}\]

  5. Simplified0.1

    \[\leadsto x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + \color{blue}{\mathsf{fma}\left(y, 5.0, x \cdot t\right)}\]
  6. Using strategy rm

  7. Applied fma-udef0.1

    \[\leadsto x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + \color{blue}{\left(y \cdot 5.0 + x \cdot t\right)}\]

  8. Applied associate-+r+0.1

    \[\leadsto \color{blue}{\left(x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + y \cdot 5.0\right) + x \cdot t}\]

  9. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(5.0, y, x \cdot \left(z + y\right) + x \cdot \left(z + y\right)\right)} + x \cdot t\]

  10. Final simplification0.0

    \[\leadsto x \cdot t + \mathsf{fma}\left(5.0, y, \left(y + z\right) \cdot x + \left(y + z\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))