Average Error: 2.0 → 0.8
Time: 21.4s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \le -1.827665421516834 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{1}{\frac{y}{z \cdot x}} + t\right) - \frac{t \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \le -1.1400369656386411 \cdot 10^{-181}:\\ \;\;\;\;\left(z \cdot \frac{x}{y} - \frac{t}{\frac{y}{x}}\right) + t\\ \mathbf{elif}\;\frac{x}{y} \le 0.0:\\ \;\;\;\;\left(\frac{1}{\frac{y}{z \cdot x}} + t\right) - \frac{t \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \le 1.0213694029549854 \cdot 10^{+99}:\\ \;\;\;\;\left(z \cdot \frac{x}{y} - \frac{t}{\frac{y}{x}}\right) + t\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z - t}{y} \cdot x\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{y} \le -1.827665421516834 \cdot 10^{+160}:\\
\;\;\;\;\left(\frac{1}{\frac{y}{z \cdot x}} + t\right) - \frac{t \cdot x}{y}\\

\mathbf{elif}\;\frac{x}{y} \le -1.1400369656386411 \cdot 10^{-181}:\\
\;\;\;\;\left(z \cdot \frac{x}{y} - \frac{t}{\frac{y}{x}}\right) + t\\

\mathbf{elif}\;\frac{x}{y} \le 0.0:\\
\;\;\;\;\left(\frac{1}{\frac{y}{z \cdot x}} + t\right) - \frac{t \cdot x}{y}\\

\mathbf{elif}\;\frac{x}{y} \le 1.0213694029549854 \cdot 10^{+99}:\\
\;\;\;\;\left(z \cdot \frac{x}{y} - \frac{t}{\frac{y}{x}}\right) + t\\

\mathbf{else}:\\
\;\;\;\;t + \frac{z - t}{y} \cdot x\\

\end{array}
double f(double x, double y, double z, double t) {
        double r23107389 = x;
        double r23107390 = y;
        double r23107391 = r23107389 / r23107390;
        double r23107392 = z;
        double r23107393 = t;
        double r23107394 = r23107392 - r23107393;
        double r23107395 = r23107391 * r23107394;
        double r23107396 = r23107395 + r23107393;
        return r23107396;
}
↓
double f(double x, double y, double z, double t) {
        double r23107397 = x;
        double r23107398 = y;
        double r23107399 = r23107397 / r23107398;
        double r23107400 = -1.827665421516834e+160;
        bool r23107401 = r23107399 <= r23107400;
        double r23107402 = 1.0;
        double r23107403 = z;
        double r23107404 = r23107403 * r23107397;
        double r23107405 = r23107398 / r23107404;
        double r23107406 = r23107402 / r23107405;
        double r23107407 = t;
        double r23107408 = r23107406 + r23107407;
        double r23107409 = r23107407 * r23107397;
        double r23107410 = r23107409 / r23107398;
        double r23107411 = r23107408 - r23107410;
        double r23107412 = -1.1400369656386411e-181;
        bool r23107413 = r23107399 <= r23107412;
        double r23107414 = r23107403 * r23107399;
        double r23107415 = r23107398 / r23107397;
        double r23107416 = r23107407 / r23107415;
        double r23107417 = r23107414 - r23107416;
        double r23107418 = r23107417 + r23107407;
        double r23107419 = 0.0;
        bool r23107420 = r23107399 <= r23107419;
        double r23107421 = 1.0213694029549854e+99;
        bool r23107422 = r23107399 <= r23107421;
        double r23107423 = r23107403 - r23107407;
        double r23107424 = r23107423 / r23107398;
        double r23107425 = r23107424 * r23107397;
        double r23107426 = r23107407 + r23107425;
        double r23107427 = r23107422 ? r23107418 : r23107426;
        double r23107428 = r23107420 ? r23107411 : r23107427;
        double r23107429 = r23107413 ? r23107418 : r23107428;
        double r23107430 = r23107401 ? r23107411 : r23107429;
        return r23107430;
}

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

Original2.0
Target2.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ x y) < -1.827665421516834e+160 or -1.1400369656386411e-181 < (/ x y) < 0.0

    1. Initial program 3.8

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

    2. Taylor expanded around 0 0.7

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

    4. Applied clear-num0.8

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

    if -1.827665421516834e+160 < (/ x y) < -1.1400369656386411e-181 or 0.0 < (/ x y) < 1.0213694029549854e+99

    1. Initial program 0.2

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

    2. Taylor expanded around 0 8.4

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

    4. Applied clear-num8.4

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

    6. Applied div-inv8.5

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

    7. Applied add-cube-cbrt8.5

      \[\leadsto \left(t + \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{y \cdot \frac{1}{x \cdot z}}\right) - \frac{t \cdot x}{y}\]

    8. Applied times-frac8.5

      \[\leadsto \left(t + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x \cdot z}}}\right) - \frac{t \cdot x}{y}\]

    9. Simplified8.5

      \[\leadsto \left(t + \color{blue}{\frac{1}{y}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x \cdot z}}\right) - \frac{t \cdot x}{y}\]

    10. Simplified8.4

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

    12. Applied associate--l+8.4

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

    13. Simplified0.2

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

    if 1.0213694029549854e+99 < (/ x y)

    1. Initial program 9.4

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

    3. Applied div-inv9.5

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

    4. Applied associate-*l*5.1

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

    5. Simplified5.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \le -1.827665421516834 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{1}{\frac{y}{z \cdot x}} + t\right) - \frac{t \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \le -1.1400369656386411 \cdot 10^{-181}:\\ \;\;\;\;\left(z \cdot \frac{x}{y} - \frac{t}{\frac{y}{x}}\right) + t\\ \mathbf{elif}\;\frac{x}{y} \le 0.0:\\ \;\;\;\;\left(\frac{1}{\frac{y}{z \cdot x}} + t\right) - \frac{t \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \le 1.0213694029549854 \cdot 10^{+99}:\\ \;\;\;\;\left(z \cdot \frac{x}{y} - \frac{t}{\frac{y}{x}}\right) + t\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z - t}{y} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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