Average Error: 6.8 → 2.1
Time: 11.0s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.569505825407989543812372125064038589152 \cdot 10^{-260}:\\ \;\;\;\;x - \frac{1}{\frac{1}{\frac{x - y}{\frac{t}{z}}}}\\ \mathbf{elif}\;x \le 5.917060633856001102745021389922587405998 \cdot 10^{-143}:\\ \;\;\;\;x - \frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x - y\right) \cdot \frac{z}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.569505825407989543812372125064038589152 \cdot 10^{-260}:\\
\;\;\;\;x - \frac{1}{\frac{1}{\frac{x - y}{\frac{t}{z}}}}\\

\mathbf{elif}\;x \le 5.917060633856001102745021389922587405998 \cdot 10^{-143}:\\
\;\;\;\;x - \frac{z \cdot \left(x - y\right)}{t}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r321243 = x;
        double r321244 = y;
        double r321245 = r321244 - r321243;
        double r321246 = z;
        double r321247 = r321245 * r321246;
        double r321248 = t;
        double r321249 = r321247 / r321248;
        double r321250 = r321243 + r321249;
        return r321250;
}
↓
double f(double x, double y, double z, double t) {
        double r321251 = x;
        double r321252 = -8.56950582540799e-260;
        bool r321253 = r321251 <= r321252;
        double r321254 = 1.0;
        double r321255 = y;
        double r321256 = r321251 - r321255;
        double r321257 = t;
        double r321258 = z;
        double r321259 = r321257 / r321258;
        double r321260 = r321256 / r321259;
        double r321261 = r321254 / r321260;
        double r321262 = r321254 / r321261;
        double r321263 = r321251 - r321262;
        double r321264 = 5.917060633856001e-143;
        bool r321265 = r321251 <= r321264;
        double r321266 = r321258 * r321256;
        double r321267 = r321266 / r321257;
        double r321268 = r321251 - r321267;
        double r321269 = r321258 / r321257;
        double r321270 = r321256 * r321269;
        double r321271 = r321251 - r321270;
        double r321272 = r321265 ? r321268 : r321271;
        double r321273 = r321253 ? r321263 : r321272;
        return r321273;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.8
Target2.0
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -8.56950582540799e-260

    1. Initial program 7.0

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

    2. Simplified1.8

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

    4. Applied *-un-lft-identity1.8

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

    5. Applied associate-*l*1.8

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

    6. Simplified1.7

      \[\leadsto x - 1 \cdot \color{blue}{\frac{x - y}{\frac{t}{z}}}\]
    7. Using strategy rm

    8. Applied div-inv1.7

      \[\leadsto x - 1 \cdot \frac{x - y}{\color{blue}{t \cdot \frac{1}{z}}}\]

    9. Applied associate-/r*7.3

      \[\leadsto x - 1 \cdot \color{blue}{\frac{\frac{x - y}{t}}{\frac{1}{z}}}\]
    10. Using strategy rm

    11. Applied clear-num7.3

      \[\leadsto x - 1 \cdot \color{blue}{\frac{1}{\frac{\frac{1}{z}}{\frac{x - y}{t}}}}\]

    12. Simplified1.7

      \[\leadsto x - 1 \cdot \frac{1}{\color{blue}{\frac{1}{\frac{x - y}{\frac{t}{z}}}}}\]

    if -8.56950582540799e-260 < x < 5.917060633856001e-143

    1. Initial program 5.6

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

    2. Simplified4.9

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

    4. Applied *-un-lft-identity4.9

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

    5. Applied associate-*l*4.9

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

    6. Simplified5.0

      \[\leadsto x - 1 \cdot \color{blue}{\frac{x - y}{\frac{t}{z}}}\]

    7. Taylor expanded around 0 5.6

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

    8. Simplified5.6

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

    if 5.917060633856001e-143 < x

    1. Initial program 7.1

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

    2. Simplified0.7

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.569505825407989543812372125064038589152 \cdot 10^{-260}:\\ \;\;\;\;x - \frac{1}{\frac{1}{\frac{x - y}{\frac{t}{z}}}}\\ \mathbf{elif}\;x \le 5.917060633856001102745021389922587405998 \cdot 10^{-143}:\\ \;\;\;\;x - \frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x - y\right) \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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