Average Error: 23.5 → 8.9
Time: 33.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.1392530337654849 \cdot 10^{-282}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, y - x, x\right)\\ \end{array}\]
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 -1.1392530337654849 \cdot 10^{-282}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, y - x, x\right)\\

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

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, y - x, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r25825452 = x;
        double r25825453 = y;
        double r25825454 = r25825453 - r25825452;
        double r25825455 = z;
        double r25825456 = t;
        double r25825457 = r25825455 - r25825456;
        double r25825458 = r25825454 * r25825457;
        double r25825459 = a;
        double r25825460 = r25825459 - r25825456;
        double r25825461 = r25825458 / r25825460;
        double r25825462 = r25825452 + r25825461;
        return r25825462;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25825463 = x;
        double r25825464 = y;
        double r25825465 = r25825464 - r25825463;
        double r25825466 = z;
        double r25825467 = t;
        double r25825468 = r25825466 - r25825467;
        double r25825469 = r25825465 * r25825468;
        double r25825470 = a;
        double r25825471 = r25825470 - r25825467;
        double r25825472 = r25825469 / r25825471;
        double r25825473 = r25825463 + r25825472;
        double r25825474 = -1.1392530337654849e-282;
        bool r25825475 = r25825473 <= r25825474;
        double r25825476 = r25825466 / r25825471;
        double r25825477 = 1.0;
        double r25825478 = r25825471 / r25825467;
        double r25825479 = r25825477 / r25825478;
        double r25825480 = r25825476 - r25825479;
        double r25825481 = fma(r25825480, r25825465, r25825463);
        double r25825482 = 0.0;
        bool r25825483 = r25825473 <= r25825482;
        double r25825484 = r25825463 / r25825467;
        double r25825485 = fma(r25825484, r25825466, r25825464);
        double r25825486 = r25825466 / r25825467;
        double r25825487 = r25825464 * r25825486;
        double r25825488 = r25825485 - r25825487;
        double r25825489 = r25825483 ? r25825488 : r25825481;
        double r25825490 = r25825475 ? r25825481 : r25825489;
        return r25825490;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original23.5
Target9.4
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.1392530337654849e-282 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 20.1

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

    2. Simplified7.5

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

    4. Applied div-sub7.5

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

    6. Applied clear-num7.6

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

    if -1.1392530337654849e-282 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

    1. Initial program 58.8

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

    2. Simplified58.8

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

    3. Taylor expanded around inf 19.8

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

    4. Simplified22.2

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.1392530337654849 \cdot 10^{-282}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, y - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))