Average Error: 1.5 → 1.5
Time: 25.7s
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 r20992376 = x;
        double r20992377 = y;
        double r20992378 = z;
        double r20992379 = t;
        double r20992380 = r20992378 - r20992379;
        double r20992381 = a;
        double r20992382 = r20992381 - r20992379;
        double r20992383 = r20992380 / r20992382;
        double r20992384 = r20992377 * r20992383;
        double r20992385 = r20992376 + r20992384;
        return r20992385;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r20992386 = z;
        double r20992387 = t;
        double r20992388 = r20992386 - r20992387;
        double r20992389 = a;
        double r20992390 = r20992389 - r20992387;
        double r20992391 = r20992388 / r20992390;
        double r20992392 = y;
        double r20992393 = x;
        double r20992394 = fma(r20992391, r20992392, r20992393);
        return r20992394;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.5
Target0.5
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  3. Final simplification1.5

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

Reproduce

herbie shell --seed 2019172 +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)))))