Average Error: 10.7 → 0.9
Time: 4.0s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -8.9131196988724503 \cdot 10^{-21} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 2.94945382370982372 \cdot 10^{299}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -8.9131196988724503 \cdot 10^{-21} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 2.94945382370982372 \cdot 10^{299}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{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 r349460 = x;
        double r349461 = y;
        double r349462 = z;
        double r349463 = r349461 - r349462;
        double r349464 = t;
        double r349465 = r349463 * r349464;
        double r349466 = a;
        double r349467 = r349466 - r349462;
        double r349468 = r349465 / r349467;
        double r349469 = r349460 + r349468;
        return r349469;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r349470 = y;
        double r349471 = z;
        double r349472 = r349470 - r349471;
        double r349473 = t;
        double r349474 = r349472 * r349473;
        double r349475 = a;
        double r349476 = r349475 - r349471;
        double r349477 = r349474 / r349476;
        double r349478 = -8.91311969887245e-21;
        bool r349479 = r349477 <= r349478;
        double r349480 = 2.9494538237098237e+299;
        bool r349481 = r349477 <= r349480;
        double r349482 = !r349481;
        bool r349483 = r349479 || r349482;
        double r349484 = r349470 / r349476;
        double r349485 = r349471 / r349476;
        double r349486 = r349484 - r349485;
        double r349487 = x;
        double r349488 = fma(r349486, r349473, r349487);
        double r349489 = r349487 + r349477;
        double r349490 = r349483 ? r349488 : r349489;
        return r349490;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.7
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- y z) t) (- a z)) < -8.91311969887245e-21 or 2.9494538237098237e+299 < (/ (* (- y z) t) (- a z))

    1. Initial program 31.4

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

    2. Simplified2.0

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

    4. Applied div-sub2.0

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

    if -8.91311969887245e-21 < (/ (* (- y z) t) (- a z)) < 2.9494538237098237e+299

    1. Initial program 0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -8.9131196988724503 \cdot 10^{-21} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 2.94945382370982372 \cdot 10^{299}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))))