Average Error: 24.3 → 10.8
Time: 15.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.415857940976969603886528580327419405542 \cdot 10^{-58} \lor \neg \left(a \le 1.530202389351854798361295608077459079059 \cdot 10^{-220}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.415857940976969603886528580327419405542 \cdot 10^{-58} \lor \neg \left(a \le 1.530202389351854798361295608077459079059 \cdot 10^{-220}\right):\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r513401 = x;
        double r513402 = y;
        double r513403 = r513402 - r513401;
        double r513404 = z;
        double r513405 = t;
        double r513406 = r513404 - r513405;
        double r513407 = r513403 * r513406;
        double r513408 = a;
        double r513409 = r513408 - r513405;
        double r513410 = r513407 / r513409;
        double r513411 = r513401 + r513410;
        return r513411;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r513412 = a;
        double r513413 = -2.4158579409769696e-58;
        bool r513414 = r513412 <= r513413;
        double r513415 = 1.5302023893518548e-220;
        bool r513416 = r513412 <= r513415;
        double r513417 = !r513416;
        bool r513418 = r513414 || r513417;
        double r513419 = y;
        double r513420 = x;
        double r513421 = r513419 - r513420;
        double r513422 = z;
        double r513423 = t;
        double r513424 = r513422 - r513423;
        double r513425 = r513412 - r513423;
        double r513426 = r513424 / r513425;
        double r513427 = fma(r513421, r513426, r513420);
        double r513428 = r513420 / r513423;
        double r513429 = r513422 * r513419;
        double r513430 = r513429 / r513423;
        double r513431 = r513419 - r513430;
        double r513432 = fma(r513428, r513422, r513431);
        double r513433 = r513418 ? r513427 : r513432;
        return r513433;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.3
Target9.3
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.4158579409769696e-58 or 1.5302023893518548e-220 < a

    1. Initial program 22.9

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

    2. Simplified11.5

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

    4. Applied fma-udef11.6

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

    6. Applied div-inv11.6

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

    7. Applied associate-*l*9.3

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

    8. Simplified9.3

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

    10. Applied fma-def9.2

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

    if -2.4158579409769696e-58 < a < 1.5302023893518548e-220

    1. Initial program 28.6

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

    2. Simplified24.1

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

    4. Applied fma-udef24.1

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

    6. Applied div-inv24.2

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

    7. Applied associate-*l*19.9

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

    8. Simplified19.8

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

    9. Taylor expanded around inf 16.0

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

    10. Simplified15.7

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.415857940976969603886528580327419405542 \cdot 10^{-58} \lor \neg \left(a \le 1.530202389351854798361295608077459079059 \cdot 10^{-220}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :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))))