Average Error: 1.3 → 1.3
Time: 9.8s
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 r10679093 = x;
        double r10679094 = y;
        double r10679095 = z;
        double r10679096 = t;
        double r10679097 = r10679095 - r10679096;
        double r10679098 = a;
        double r10679099 = r10679095 - r10679098;
        double r10679100 = r10679097 / r10679099;
        double r10679101 = r10679094 * r10679100;
        double r10679102 = r10679093 + r10679101;
        return r10679102;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r10679103 = z;
        double r10679104 = t;
        double r10679105 = r10679103 - r10679104;
        double r10679106 = a;
        double r10679107 = r10679103 - r10679106;
        double r10679108 = r10679105 / r10679107;
        double r10679109 = y;
        double r10679110 = x;
        double r10679111 = fma(r10679108, r10679109, r10679110);
        return r10679111;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target1.2
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 div-inv1.4

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

  6. Applied un-div-inv1.3

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

  7. Final simplification1.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"

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

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