Average Error: 11.0 → 0.3
Time: 6.2s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 2.52318601275925725 \cdot 10^{284}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]
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 2.52318601275925725 \cdot 10^{284}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r646325 = x;
        double r646326 = y;
        double r646327 = z;
        double r646328 = r646326 - r646327;
        double r646329 = t;
        double r646330 = r646328 * r646329;
        double r646331 = a;
        double r646332 = r646331 - r646327;
        double r646333 = r646330 / r646332;
        double r646334 = r646325 + r646333;
        return r646334;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r646335 = y;
        double r646336 = z;
        double r646337 = r646335 - r646336;
        double r646338 = t;
        double r646339 = r646337 * r646338;
        double r646340 = a;
        double r646341 = r646340 - r646336;
        double r646342 = r646339 / r646341;
        double r646343 = -inf.0;
        bool r646344 = r646342 <= r646343;
        double r646345 = 2.5231860127592572e+284;
        bool r646346 = r646342 <= r646345;
        double r646347 = !r646346;
        bool r646348 = r646344 || r646347;
        double r646349 = r646337 / r646341;
        double r646350 = x;
        double r646351 = fma(r646349, r646338, r646350);
        double r646352 = r646350 + r646342;
        double r646353 = r646348 ? r646351 : r646352;
        return r646353;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target0.6
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 2.5231860127592572e+284 < (/ (* (- y z) t) (- a z))

    1. Initial program 62.4

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

    2. Simplified0.4

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

    if -inf.0 < (/ (* (- y z) t) (- a z)) < 2.5231860127592572e+284

    1. Initial program 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 2.52318601275925725 \cdot 10^{284}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))