Average Error: 1.4 → 1.5
Time: 4.4s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{z - a}, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{z - a}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r601906 = x;
        double r601907 = y;
        double r601908 = z;
        double r601909 = t;
        double r601910 = r601908 - r601909;
        double r601911 = a;
        double r601912 = r601908 - r601911;
        double r601913 = r601910 / r601912;
        double r601914 = r601907 * r601913;
        double r601915 = r601906 + r601914;
        return r601915;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r601916 = y;
        double r601917 = z;
        double r601918 = t;
        double r601919 = r601917 - r601918;
        double r601920 = 1.0;
        double r601921 = a;
        double r601922 = r601917 - r601921;
        double r601923 = r601920 / r601922;
        double r601924 = r601919 * r601923;
        double r601925 = x;
        double r601926 = fma(r601916, r601924, r601925);
        return r601926;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target1.3
Herbie1.5
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  4. Applied div-inv1.5

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

  5. Final simplification1.5

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

Reproduce

herbie shell --seed 2020001 +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)))))