Average Error: 1.5 → 1.5
Time: 21.5s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{a - t}}{\frac{1}{z - t}}, y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\frac{\frac{1}{a - t}}{\frac{1}{z - t}}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r385345 = x;
        double r385346 = y;
        double r385347 = z;
        double r385348 = t;
        double r385349 = r385347 - r385348;
        double r385350 = a;
        double r385351 = r385350 - r385348;
        double r385352 = r385349 / r385351;
        double r385353 = r385346 * r385352;
        double r385354 = r385345 + r385353;
        return r385354;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r385355 = 1.0;
        double r385356 = a;
        double r385357 = t;
        double r385358 = r385356 - r385357;
        double r385359 = r385355 / r385358;
        double r385360 = z;
        double r385361 = r385360 - r385357;
        double r385362 = r385355 / r385361;
        double r385363 = r385359 / r385362;
        double r385364 = y;
        double r385365 = x;
        double r385366 = fma(r385363, r385364, r385365);
        return r385366;
}

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. Using strategy rm

  4. Applied clear-num1.6

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

  6. Applied div-inv1.6

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

  7. Applied associate-/r*1.5

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

  8. Final simplification1.5

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.50808486055124107e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.8944268627920891e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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