Average Error: 11.0 → 0.3
Time: 4.5s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 7.1006285597022056 \cdot 10^{249}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{y - z}}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(y - z\right) \cdot t}{a - z} + x\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 7.1006285597022056 \cdot 10^{249}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{y - z}}, t, x\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \frac{\left(y - z\right) \cdot t}{a - z} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r607271 = x;
        double r607272 = y;
        double r607273 = z;
        double r607274 = r607272 - r607273;
        double r607275 = t;
        double r607276 = r607274 * r607275;
        double r607277 = a;
        double r607278 = r607277 - r607273;
        double r607279 = r607276 / r607278;
        double r607280 = r607271 + r607279;
        return r607280;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r607281 = y;
        double r607282 = z;
        double r607283 = r607281 - r607282;
        double r607284 = t;
        double r607285 = r607283 * r607284;
        double r607286 = a;
        double r607287 = r607286 - r607282;
        double r607288 = r607285 / r607287;
        double r607289 = -inf.0;
        bool r607290 = r607288 <= r607289;
        double r607291 = 7.100628559702206e+249;
        bool r607292 = r607288 <= r607291;
        double r607293 = !r607292;
        bool r607294 = r607290 || r607293;
        double r607295 = 1.0;
        double r607296 = r607287 / r607283;
        double r607297 = r607295 / r607296;
        double r607298 = x;
        double r607299 = fma(r607297, r607284, r607298);
        double r607300 = r607295 * r607288;
        double r607301 = r607300 + r607298;
        double r607302 = r607294 ? r607299 : r607301;
        return r607302;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target0.7
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- y z) t) (- a z)) < -inf.0 or 7.100628559702206e+249 < (/ (* (- y z) t) (- a z))

    1. Initial program 59.7

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

    2. Simplified0.8

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

    4. Applied clear-num0.9

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

    if -inf.0 < (/ (* (- y z) t) (- a z)) < 7.100628559702206e+249

    1. Initial program 0.2

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

    2. Simplified1.6

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

    4. Applied clear-num1.6

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

    6. Applied fma-udef1.6

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

    7. Simplified3.1

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

    9. Applied *-un-lft-identity3.1

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

    10. Applied associate-*l*3.1

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

    11. Simplified0.2

      \[\leadsto 1 \cdot \color{blue}{\frac{\left(y - z\right) \cdot t}{a - z}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 7.1006285597022056 \cdot 10^{249}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{y - z}}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(y - z\right) \cdot t}{a - z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))