Average Error: 10.9 → 1.2
Time: 8.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[y \cdot \frac{z - t}{a - t} + x\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
y \cdot \frac{z - t}{a - t} + x
double f(double x, double y, double z, double t, double a) {
        double r667365 = x;
        double r667366 = y;
        double r667367 = z;
        double r667368 = t;
        double r667369 = r667367 - r667368;
        double r667370 = r667366 * r667369;
        double r667371 = a;
        double r667372 = r667371 - r667368;
        double r667373 = r667370 / r667372;
        double r667374 = r667365 + r667373;
        return r667374;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r667375 = y;
        double r667376 = z;
        double r667377 = t;
        double r667378 = r667376 - r667377;
        double r667379 = a;
        double r667380 = r667379 - r667377;
        double r667381 = r667378 / r667380;
        double r667382 = r667375 * r667381;
        double r667383 = x;
        double r667384 = r667382 + r667383;
        return r667384;
}

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

Original10.9
Target1.0
Herbie1.2
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 10.9

    \[x + \frac{y \cdot \left(z - t\right)}{a - t}\]

  2. Simplified3.0

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

  4. Applied fma-udef3.0

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

  6. Applied div-inv3.0

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

  7. Applied associate-*l*1.2

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

  8. Simplified1.2

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

  9. Final simplification1.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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