Average Error: 10.4 → 1.3
Time: 11.7s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 113823952470359359488:\\ \;\;\;\;t \cdot \frac{y - z}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t} - \frac{z}{t}}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 113823952470359359488:\\
\;\;\;\;t \cdot \frac{y - z}{a - z} + x\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y - z}{\frac{a}{t} - \frac{z}{t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r421290 = x;
        double r421291 = y;
        double r421292 = z;
        double r421293 = r421291 - r421292;
        double r421294 = t;
        double r421295 = r421293 * r421294;
        double r421296 = a;
        double r421297 = r421296 - r421292;
        double r421298 = r421295 / r421297;
        double r421299 = r421290 + r421298;
        return r421299;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r421300 = y;
        double r421301 = z;
        double r421302 = r421300 - r421301;
        double r421303 = t;
        double r421304 = r421302 * r421303;
        double r421305 = a;
        double r421306 = r421305 - r421301;
        double r421307 = r421304 / r421306;
        double r421308 = 1.1382395247035936e+20;
        bool r421309 = r421307 <= r421308;
        double r421310 = r421302 / r421306;
        double r421311 = r421303 * r421310;
        double r421312 = x;
        double r421313 = r421311 + r421312;
        double r421314 = r421305 / r421303;
        double r421315 = r421301 / r421303;
        double r421316 = r421314 - r421315;
        double r421317 = r421302 / r421316;
        double r421318 = r421312 + r421317;
        double r421319 = r421309 ? r421313 : r421318;
        return r421319;
}

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

Original10.4
Target0.6
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- y z) t) (- a z)) < 1.1382395247035936e+20

    1. Initial program 7.0

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

    3. Applied associate-/l*3.3

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

    5. Applied *-un-lft-identity3.3

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

    6. Applied *-un-lft-identity3.3

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

    7. Applied times-frac3.3

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

    8. Applied *-un-lft-identity3.3

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

    9. Applied times-frac3.3

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

    10. Simplified3.3

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

    11. Simplified0.9

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

    if 1.1382395247035936e+20 < (/ (* (- y z) t) (- a z))

    1. Initial program 24.2

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

    3. Applied associate-/l*2.5

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

    5. Applied div-sub2.5

      \[\leadsto x + \frac{y - z}{\color{blue}{\frac{a}{t} - \frac{z}{t}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 113823952470359359488:\\ \;\;\;\;t \cdot \frac{y - z}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t} - \frac{z}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))