Average Error: 1.5 → 1.5
Time: 20.9s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{-1 + \frac{a}{t}}, y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{-1 + \frac{a}{t}}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r356233 = x;
        double r356234 = y;
        double r356235 = z;
        double r356236 = t;
        double r356237 = r356235 - r356236;
        double r356238 = a;
        double r356239 = r356238 - r356236;
        double r356240 = r356237 / r356239;
        double r356241 = r356234 * r356240;
        double r356242 = r356233 + r356241;
        return r356242;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r356243 = z;
        double r356244 = a;
        double r356245 = t;
        double r356246 = r356244 - r356245;
        double r356247 = r356243 / r356246;
        double r356248 = 1.0;
        double r356249 = -1.0;
        double r356250 = r356244 / r356245;
        double r356251 = r356249 + r356250;
        double r356252 = r356248 / r356251;
        double r356253 = r356247 - r356252;
        double r356254 = y;
        double r356255 = x;
        double r356256 = fma(r356253, r356254, r356255);
        return r356256;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.5
Target0.4
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 div-sub1.5

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

  6. Applied clear-num1.5

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

  7. Simplified1.5

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

  8. Final simplification1.5

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

Reproduce

herbie shell --seed 2019174 +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)))))