Average Error: 24.3 → 9.6
Time: 19.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -8.572794558051977100123683312677994148346 \cdot 10^{-160} \lor \neg \left(a \le 1.021106734991105879334115178573731189003 \cdot 10^{-168}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{z - t} - \frac{\sqrt[3]{t}}{\sqrt[3]{z - t}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \sqrt[3]{t}\right)}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - \frac{z}{t} \cdot y\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -8.572794558051977100123683312677994148346 \cdot 10^{-160} \lor \neg \left(a \le 1.021106734991105879334115178573731189003 \cdot 10^{-168}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{z - t} - \frac{\sqrt[3]{t}}{\sqrt[3]{z - t}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \sqrt[3]{t}\right)}, y - x, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r414393 = x;
        double r414394 = y;
        double r414395 = r414394 - r414393;
        double r414396 = z;
        double r414397 = t;
        double r414398 = r414396 - r414397;
        double r414399 = r414395 * r414398;
        double r414400 = a;
        double r414401 = r414400 - r414397;
        double r414402 = r414399 / r414401;
        double r414403 = r414393 + r414402;
        return r414403;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r414404 = a;
        double r414405 = -8.572794558051977e-160;
        bool r414406 = r414404 <= r414405;
        double r414407 = 1.0211067349911059e-168;
        bool r414408 = r414404 <= r414407;
        double r414409 = !r414408;
        bool r414410 = r414406 || r414409;
        double r414411 = 1.0;
        double r414412 = z;
        double r414413 = t;
        double r414414 = r414412 - r414413;
        double r414415 = r414404 / r414414;
        double r414416 = cbrt(r414413);
        double r414417 = cbrt(r414414);
        double r414418 = r414416 / r414417;
        double r414419 = r414417 * r414417;
        double r414420 = r414416 / r414419;
        double r414421 = r414420 * r414416;
        double r414422 = r414418 * r414421;
        double r414423 = r414415 - r414422;
        double r414424 = r414411 / r414423;
        double r414425 = y;
        double r414426 = x;
        double r414427 = r414425 - r414426;
        double r414428 = fma(r414424, r414427, r414426);
        double r414429 = r414426 / r414413;
        double r414430 = fma(r414429, r414412, r414425);
        double r414431 = r414412 / r414413;
        double r414432 = r414431 * r414425;
        double r414433 = r414430 - r414432;
        double r414434 = r414410 ? r414428 : r414433;
        return r414434;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.3
Target9.2
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -8.572794558051977e-160 or 1.0211067349911059e-168 < a

    1. Initial program 23.0

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

    2. Simplified9.3

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

    4. Applied clear-num9.4

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

    6. Applied div-sub9.4

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

    8. Applied add-cube-cbrt9.7

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

    9. Applied add-cube-cbrt9.5

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

    10. Applied times-frac9.5

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

    11. Simplified9.5

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

    if -8.572794558051977e-160 < a < 1.0211067349911059e-168

    1. Initial program 29.5

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

    2. Simplified19.8

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

    3. Taylor expanded around inf 12.5

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

    4. Simplified10.1

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.572794558051977100123683312677994148346 \cdot 10^{-160} \lor \neg \left(a \le 1.021106734991105879334115178573731189003 \cdot 10^{-168}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{z - t} - \frac{\sqrt[3]{t}}{\sqrt[3]{z - t}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \sqrt[3]{t}\right)}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - \frac{z}{t} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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