Average Error: 24.5 → 8.6
Time: 52.7s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.490239212793318560011781773848949760307 \cdot 10^{-243}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot \left(y - x\right) + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot \left(y - x\right) + x\\ \end{array}\]
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 -1.490239212793318560011781773848949760307 \cdot 10^{-243}:\\
\;\;\;\;\frac{z - t}{a - t} \cdot \left(y - x\right) + x\\

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

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r116531370 = x;
        double r116531371 = y;
        double r116531372 = r116531371 - r116531370;
        double r116531373 = z;
        double r116531374 = t;
        double r116531375 = r116531373 - r116531374;
        double r116531376 = r116531372 * r116531375;
        double r116531377 = a;
        double r116531378 = r116531377 - r116531374;
        double r116531379 = r116531376 / r116531378;
        double r116531380 = r116531370 + r116531379;
        return r116531380;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r116531381 = x;
        double r116531382 = y;
        double r116531383 = r116531382 - r116531381;
        double r116531384 = z;
        double r116531385 = t;
        double r116531386 = r116531384 - r116531385;
        double r116531387 = r116531383 * r116531386;
        double r116531388 = a;
        double r116531389 = r116531388 - r116531385;
        double r116531390 = r116531387 / r116531389;
        double r116531391 = r116531381 + r116531390;
        double r116531392 = -1.4902392127933186e-243;
        bool r116531393 = r116531391 <= r116531392;
        double r116531394 = r116531386 / r116531389;
        double r116531395 = r116531394 * r116531383;
        double r116531396 = r116531395 + r116531381;
        double r116531397 = 0.0;
        bool r116531398 = r116531391 <= r116531397;
        double r116531399 = r116531381 * r116531384;
        double r116531400 = r116531399 / r116531385;
        double r116531401 = r116531382 + r116531400;
        double r116531402 = r116531384 * r116531382;
        double r116531403 = r116531402 / r116531385;
        double r116531404 = r116531401 - r116531403;
        double r116531405 = r116531398 ? r116531404 : r116531396;
        double r116531406 = r116531393 ? r116531396 : r116531405;
        return r116531406;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.5
Target9.3
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.4902392127933186e-243 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 21.4

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

    2. Simplified10.1

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

    4. Applied div-sub10.1

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

    6. Applied div-inv10.1

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

    8. Applied fma-udef10.1

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

    9. Simplified10.1

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

    11. Applied div-inv10.1

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

    12. Applied div-inv10.2

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

    13. Applied distribute-rgt-out--10.2

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

    14. Applied associate-*r*7.4

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

    15. Simplified7.3

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

    if -1.4902392127933186e-243 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

    1. Initial program 55.5

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

    2. Simplified55.9

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

    4. Applied div-sub55.9

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

    6. Applied div-inv56.0

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

    8. Applied fma-udef56.1

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

    9. Simplified56.2

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

    10. Taylor expanded around inf 21.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.490239212793318560011781773848949760307 \cdot 10^{-243}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot \left(y - x\right) + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot \left(y - x\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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))))