Average Error: 11.1 → 1.5
Time: 4.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\frac{\frac{z - t}{a - t}}{\frac{1}{y}} + x\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\frac{\frac{z - t}{a - t}}{\frac{1}{y}} + x
double f(double x, double y, double z, double t, double a) {
        double r669393 = x;
        double r669394 = y;
        double r669395 = z;
        double r669396 = t;
        double r669397 = r669395 - r669396;
        double r669398 = r669394 * r669397;
        double r669399 = a;
        double r669400 = r669399 - r669396;
        double r669401 = r669398 / r669400;
        double r669402 = r669393 + r669401;
        return r669402;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r669403 = z;
        double r669404 = t;
        double r669405 = r669403 - r669404;
        double r669406 = a;
        double r669407 = r669406 - r669404;
        double r669408 = r669405 / r669407;
        double r669409 = 1.0;
        double r669410 = y;
        double r669411 = r669409 / r669410;
        double r669412 = r669408 / r669411;
        double r669413 = x;
        double r669414 = r669412 + r669413;
        return r669414;
}

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

Original11.1
Target1.3
Herbie1.5
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 11.1

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

  2. Simplified3.1

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

  4. Applied clear-num3.4

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

  6. Applied fma-udef3.4

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

  7. Simplified3.2

    \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y}}} + x\]
  8. Using strategy rm

  9. Applied div-inv3.2

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

  10. Applied associate-/r*1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2020083 +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))))