Average Error: 1.3 → 1.3
Time: 47.7s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r27173142 = x;
        double r27173143 = y;
        double r27173144 = z;
        double r27173145 = t;
        double r27173146 = r27173144 - r27173145;
        double r27173147 = a;
        double r27173148 = r27173147 - r27173145;
        double r27173149 = r27173146 / r27173148;
        double r27173150 = r27173143 * r27173149;
        double r27173151 = r27173142 + r27173150;
        return r27173151;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r27173152 = z;
        double r27173153 = t;
        double r27173154 = r27173152 - r27173153;
        double r27173155 = a;
        double r27173156 = r27173155 - r27173153;
        double r27173157 = r27173154 / r27173156;
        double r27173158 = y;
        double r27173159 = x;
        double r27173160 = fma(r27173157, r27173158, r27173159);
        return r27173160;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.4
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1.0}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  3. Final simplification1.3

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

Reproduce

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

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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