Average Error: 1.3 → 1.3
Time: 19.2s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r294788 = x;
        double r294789 = y;
        double r294790 = z;
        double r294791 = t;
        double r294792 = r294790 - r294791;
        double r294793 = a;
        double r294794 = r294790 - r294793;
        double r294795 = r294792 / r294794;
        double r294796 = r294789 * r294795;
        double r294797 = r294788 + r294796;
        return r294797;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r294798 = z;
        double r294799 = t;
        double r294800 = r294798 - r294799;
        double r294801 = a;
        double r294802 = r294798 - r294801;
        double r294803 = r294800 / r294802;
        double r294804 = y;
        double r294805 = x;
        double r294806 = fma(r294803, r294804, r294805);
        return r294806;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target1.1
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.3

    \[x + y \cdot \frac{z - t}{z - a}\]

  2. Simplified1.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity1.3

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

  5. Applied *-un-lft-identity1.3

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

  6. Applied times-frac1.3

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

  7. Simplified1.3

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

  8. Final simplification1.3

    \[\leadsto \mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (* y (/ (- z t) (- z a)))))