Average Error: 10.8 → 1.5
Time: 5.0s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\frac{y - z}{a - z} \cdot t + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\frac{y - z}{a - z} \cdot t + x
double f(double x, double y, double z, double t, double a) {
        double r665736 = x;
        double r665737 = y;
        double r665738 = z;
        double r665739 = r665737 - r665738;
        double r665740 = t;
        double r665741 = r665739 * r665740;
        double r665742 = a;
        double r665743 = r665742 - r665738;
        double r665744 = r665741 / r665743;
        double r665745 = r665736 + r665744;
        return r665745;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r665746 = y;
        double r665747 = z;
        double r665748 = r665746 - r665747;
        double r665749 = a;
        double r665750 = r665749 - r665747;
        double r665751 = r665748 / r665750;
        double r665752 = t;
        double r665753 = r665751 * r665752;
        double r665754 = x;
        double r665755 = r665753 + r665754;
        return r665755;
}

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

Original10.8
Target0.6
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 10.8

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

  2. Simplified1.5

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

  4. Applied fma-udef1.5

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

  5. Final simplification1.5

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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