Average Error: 23.9 → 10.5
Time: 21.3s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.423345970921716 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \le 1.6895069144640092 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.423345970921716 \cdot 10^{+250}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\

\mathbf{elif}\;t \le 1.6895069144640092 \cdot 10^{+105}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r28240452 = x;
        double r28240453 = y;
        double r28240454 = r28240453 - r28240452;
        double r28240455 = z;
        double r28240456 = t;
        double r28240457 = r28240455 - r28240456;
        double r28240458 = r28240454 * r28240457;
        double r28240459 = a;
        double r28240460 = r28240459 - r28240456;
        double r28240461 = r28240458 / r28240460;
        double r28240462 = r28240452 + r28240461;
        return r28240462;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r28240463 = t;
        double r28240464 = -7.423345970921716e+250;
        bool r28240465 = r28240463 <= r28240464;
        double r28240466 = x;
        double r28240467 = r28240466 / r28240463;
        double r28240468 = z;
        double r28240469 = y;
        double r28240470 = fma(r28240467, r28240468, r28240469);
        double r28240471 = r28240468 / r28240463;
        double r28240472 = r28240469 * r28240471;
        double r28240473 = r28240470 - r28240472;
        double r28240474 = 1.6895069144640092e+105;
        bool r28240475 = r28240463 <= r28240474;
        double r28240476 = 1.0;
        double r28240477 = a;
        double r28240478 = r28240477 - r28240463;
        double r28240479 = r28240476 / r28240478;
        double r28240480 = r28240468 - r28240463;
        double r28240481 = r28240479 * r28240480;
        double r28240482 = r28240469 - r28240466;
        double r28240483 = fma(r28240481, r28240482, r28240466);
        double r28240484 = r28240475 ? r28240483 : r28240473;
        double r28240485 = r28240465 ? r28240473 : r28240484;
        return r28240485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original23.9
Target9.2
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.423345970921716e+250 or 1.6895069144640092e+105 < t

    1. Initial program 45.0

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

    2. Simplified22.0

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

    3. Taylor expanded around inf 25.1

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

    4. Simplified16.4

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

    if -7.423345970921716e+250 < t < 1.6895069144640092e+105

    1. Initial program 17.2

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

    2. Simplified8.6

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

    4. Applied clear-num8.7

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

    6. Applied div-inv8.7

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

    7. Applied add-cube-cbrt8.7

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(a - t\right) \cdot \frac{1}{z - t}}, y - x, x\right)\]

    8. Applied times-frac8.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a - t} \cdot \frac{\sqrt[3]{1}}{\frac{1}{z - t}}}, y - x, x\right)\]

    9. Simplified8.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{a - t}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{z - t}}, y - x, x\right)\]

    10. Simplified8.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.423345970921716 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \le 1.6895069144640092 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))