Average Error: 24.0 → 6.1
Time: 23.9s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.610319660026901692627567289598819324756 \cdot 10^{-296}:\\ \;\;\;\;\sqrt[3]{y} \cdot \left(\frac{z - t}{a - t} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(\frac{z - t}{a - t}, -x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y \cdot \frac{z - t}{a - t} + \frac{z \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{y} \cdot \left(\frac{z - t}{a - t} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(\frac{z - t}{a - t}, -x, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.610319660026901692627567289598819324756 \cdot 10^{-296}:\\
\;\;\;\;\sqrt[3]{y} \cdot \left(\frac{z - t}{a - t} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(\frac{z - t}{a - t}, -x, x\right)\\

\mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\
\;\;\;\;y \cdot \frac{z - t}{a - t} + \frac{z \cdot x}{t}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{y} \cdot \left(\frac{z - t}{a - t} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(\frac{z - t}{a - t}, -x, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r26282425 = x;
        double r26282426 = y;
        double r26282427 = r26282426 - r26282425;
        double r26282428 = z;
        double r26282429 = t;
        double r26282430 = r26282428 - r26282429;
        double r26282431 = r26282427 * r26282430;
        double r26282432 = a;
        double r26282433 = r26282432 - r26282429;
        double r26282434 = r26282431 / r26282433;
        double r26282435 = r26282425 + r26282434;
        return r26282435;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r26282436 = x;
        double r26282437 = y;
        double r26282438 = r26282437 - r26282436;
        double r26282439 = z;
        double r26282440 = t;
        double r26282441 = r26282439 - r26282440;
        double r26282442 = r26282438 * r26282441;
        double r26282443 = a;
        double r26282444 = r26282443 - r26282440;
        double r26282445 = r26282442 / r26282444;
        double r26282446 = r26282436 + r26282445;
        double r26282447 = -1.6103196600269017e-296;
        bool r26282448 = r26282446 <= r26282447;
        double r26282449 = cbrt(r26282437);
        double r26282450 = r26282441 / r26282444;
        double r26282451 = r26282449 * r26282449;
        double r26282452 = r26282450 * r26282451;
        double r26282453 = r26282449 * r26282452;
        double r26282454 = -r26282436;
        double r26282455 = fma(r26282450, r26282454, r26282436);
        double r26282456 = r26282453 + r26282455;
        double r26282457 = 0.0;
        bool r26282458 = r26282446 <= r26282457;
        double r26282459 = r26282437 * r26282450;
        double r26282460 = r26282439 * r26282436;
        double r26282461 = r26282460 / r26282440;
        double r26282462 = r26282459 + r26282461;
        double r26282463 = r26282458 ? r26282462 : r26282456;
        double r26282464 = r26282448 ? r26282456 : r26282463;
        return r26282464;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.0
Target8.5
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.6103196600269017e-296 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 20.8

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

    2. Simplified6.7

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

    4. Applied fma-udef6.7

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

    6. Applied sub-neg6.7

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

    7. Applied distribute-lft-in6.7

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

    8. Applied associate-+l+4.6

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

    9. Simplified4.6

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

    11. Applied add-cube-cbrt5.2

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

    12. Applied associate-*r*5.2

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

    if -1.6103196600269017e-296 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

    1. Initial program 60.9

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

    2. Simplified60.9

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

    4. Applied fma-udef60.9

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

    6. Applied sub-neg60.9

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

    7. Applied distribute-lft-in60.9

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

    8. Applied associate-+l+35.1

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

    9. Simplified35.1

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

    10. Taylor expanded around inf 17.0

      \[\leadsto \frac{z - t}{a - t} \cdot y + \color{blue}{\frac{x \cdot z}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.610319660026901692627567289598819324756 \cdot 10^{-296}:\\ \;\;\;\;\sqrt[3]{y} \cdot \left(\frac{z - t}{a - t} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(\frac{z - t}{a - t}, -x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y \cdot \frac{z - t}{a - t} + \frac{z \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{y} \cdot \left(\frac{z - t}{a - t} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(\frac{z - t}{a - t}, -x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))))