Average Error: 1.3 → 1.2
Time: 17.5s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\frac{y}{\frac{a - t}{z - t}} + x\]
x + y \cdot \frac{z - t}{a - t}
↓
\frac{y}{\frac{a - t}{z - t}} + x
double f(double x, double y, double z, double t, double a) {
        double r390562 = x;
        double r390563 = y;
        double r390564 = z;
        double r390565 = t;
        double r390566 = r390564 - r390565;
        double r390567 = a;
        double r390568 = r390567 - r390565;
        double r390569 = r390566 / r390568;
        double r390570 = r390563 * r390569;
        double r390571 = r390562 + r390570;
        return r390571;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r390572 = y;
        double r390573 = a;
        double r390574 = t;
        double r390575 = r390573 - r390574;
        double r390576 = z;
        double r390577 = r390576 - r390574;
        double r390578 = r390575 / r390577;
        double r390579 = r390572 / r390578;
        double r390580 = x;
        double r390581 = r390579 + r390580;
        return r390581;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.3
Target0.5
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  4. Applied clear-num1.4

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

  6. Applied fma-udef1.4

    \[\leadsto \color{blue}{\frac{1}{\frac{a - t}{z - t}} \cdot y + x}\]

  7. Simplified1.2

    \[\leadsto \color{blue}{\frac{y}{\frac{a - t}{z - t}}} + x\]

  8. Final simplification1.2

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

Reproduce

herbie shell --seed 2019303 +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)))))