Average Error: 11.0 → 0.3
Time: 7.2s
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 r647562 = x;
        double r647563 = y;
        double r647564 = z;
        double r647565 = r647563 - r647564;
        double r647566 = t;
        double r647567 = r647565 * r647566;
        double r647568 = a;
        double r647569 = r647568 - r647564;
        double r647570 = r647567 / r647569;
        double r647571 = r647562 + r647570;
        return r647571;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r647572 = y;
        double r647573 = z;
        double r647574 = r647572 - r647573;
        double r647575 = t;
        double r647576 = r647574 * r647575;
        double r647577 = a;
        double r647578 = r647577 - r647573;
        double r647579 = r647576 / r647578;
        double r647580 = -inf.0;
        bool r647581 = r647579 <= r647580;
        double r647582 = 2.5231860127592572e+284;
        bool r647583 = r647579 <= r647582;
        double r647584 = !r647583;
        bool r647585 = r647581 || r647584;
        double r647586 = r647574 / r647578;
        double r647587 = x;
        double r647588 = fma(r647586, r647575, r647587);
        double r647589 = r647587 + r647579;
        double r647590 = r647585 ? r647588 : r647589;
        return r647590;
}

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))))