Average Error: 6.5 → 1.5
Time: 12.6s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.943637715556892604205293156667117905826 \cdot 10^{-52}:\\ \;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\ \mathbf{elif}\;t \le 2.664903500369063139653685363940380050419 \cdot 10^{62}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.943637715556892604205293156667117905826 \cdot 10^{-52}:\\
\;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\

\mathbf{elif}\;t \le 2.664903500369063139653685363940380050419 \cdot 10^{62}:\\
\;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r314362 = x;
        double r314363 = y;
        double r314364 = r314363 - r314362;
        double r314365 = z;
        double r314366 = r314364 * r314365;
        double r314367 = t;
        double r314368 = r314366 / r314367;
        double r314369 = r314362 + r314368;
        return r314369;
}
↓
double f(double x, double y, double z, double t) {
        double r314370 = t;
        double r314371 = -1.9436377155568926e-52;
        bool r314372 = r314370 <= r314371;
        double r314373 = y;
        double r314374 = x;
        double r314375 = r314373 - r314374;
        double r314376 = z;
        double r314377 = r314370 / r314376;
        double r314378 = r314375 / r314377;
        double r314379 = r314378 + r314374;
        double r314380 = 2.664903500369063e+62;
        bool r314381 = r314370 <= r314380;
        double r314382 = r314375 * r314376;
        double r314383 = r314382 / r314370;
        double r314384 = r314383 + r314374;
        double r314385 = r314375 / r314370;
        double r314386 = fma(r314385, r314376, r314374);
        double r314387 = r314381 ? r314384 : r314386;
        double r314388 = r314372 ? r314379 : r314387;
        return r314388;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.5
Target2.0
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.9436377155568926e-52

    1. Initial program 8.4

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

    2. Simplified1.0

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

    4. Applied fma-udef1.0

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

    5. Simplified8.4

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

    7. Applied associate-/l*1.2

      \[\leadsto \color{blue}{\frac{y - x}{\frac{t}{z}}} + x\]

    if -1.9436377155568926e-52 < t < 2.664903500369063e+62

    1. Initial program 2.1

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

    2. Simplified13.8

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

    4. Applied fma-udef13.8

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

    5. Simplified2.1

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

    if 2.664903500369063e+62 < t

    1. Initial program 10.8

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

    2. Simplified1.0

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.943637715556892604205293156667117905826 \cdot 10^{-52}:\\ \;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\ \mathbf{elif}\;t \le 2.664903500369063139653685363940380050419 \cdot 10^{62}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.0255111955330046e-135) (- x (* (/ z t) (- x y))) (if (< x 4.2750321637007147e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))