Average Error: 24.2 → 7.4
Time: 7.3s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -8.39805018372225709 \cdot 10^{-172} \lor \neg \left(a \le 4.96130980133353584 \cdot 10^{-246}\right):\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y + \mathsf{fma}\left(-x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -8.39805018372225709 \cdot 10^{-172} \lor \neg \left(a \le 4.96130980133353584 \cdot 10^{-246}\right):\\
\;\;\;\;\left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y + \mathsf{fma}\left(-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 r908322 = x;
        double r908323 = y;
        double r908324 = r908323 - r908322;
        double r908325 = z;
        double r908326 = t;
        double r908327 = r908325 - r908326;
        double r908328 = r908324 * r908327;
        double r908329 = a;
        double r908330 = r908329 - r908326;
        double r908331 = r908328 / r908330;
        double r908332 = r908322 + r908331;
        return r908332;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r908333 = a;
        double r908334 = -8.398050183722257e-172;
        bool r908335 = r908333 <= r908334;
        double r908336 = 4.961309801333536e-246;
        bool r908337 = r908333 <= r908336;
        double r908338 = !r908337;
        bool r908339 = r908335 || r908338;
        double r908340 = z;
        double r908341 = t;
        double r908342 = r908340 - r908341;
        double r908343 = 1.0;
        double r908344 = r908333 - r908341;
        double r908345 = r908343 / r908344;
        double r908346 = r908342 * r908345;
        double r908347 = y;
        double r908348 = r908346 * r908347;
        double r908349 = x;
        double r908350 = -r908349;
        double r908351 = r908342 / r908344;
        double r908352 = fma(r908350, r908351, r908349);
        double r908353 = r908348 + r908352;
        double r908354 = r908349 / r908341;
        double r908355 = r908340 * r908347;
        double r908356 = r908355 / r908341;
        double r908357 = r908347 - r908356;
        double r908358 = fma(r908354, r908340, r908357);
        double r908359 = r908339 ? r908353 : r908358;
        return r908359;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.2
Target9.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -8.398050183722257e-172 or 4.961309801333536e-246 < a

    1. Initial program 23.1

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

    2. Simplified12.9

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

    4. Applied div-sub12.9

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

    6. Applied div-inv12.9

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

    8. Applied fma-udef13.0

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

    9. Simplified10.3

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

    11. Applied sub-neg10.3

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

    12. Applied distribute-lft-in10.3

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

    13. Applied associate-+l+7.6

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

    14. Simplified6.9

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

    if -8.398050183722257e-172 < a < 4.961309801333536e-246

    1. Initial program 31.0

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

    2. Simplified25.3

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

    4. Applied div-sub25.3

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

    6. Applied div-inv25.3

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

    8. Applied fma-udef25.3

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

    9. Simplified19.7

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

    10. Taylor expanded around inf 12.1

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

    11. Simplified10.8

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.39805018372225709 \cdot 10^{-172} \lor \neg \left(a \le 4.96130980133353584 \cdot 10^{-246}\right):\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y + \mathsf{fma}\left(-x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))))