Average Error: 2.1 → 2.5
Time: 19.4s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.664918217275974893528006684776919727564 \cdot 10^{-24}:\\ \;\;\;\;t + \frac{z - t}{\frac{y}{x}}\\ \mathbf{elif}\;t \le -1.336117363059674131707044966713149076742 \cdot 10^{-223}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{elif}\;t \le 7.81875512884679996695346065746985015342 \cdot 10^{-283}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{x}{y} + t\\ \mathbf{elif}\;t \le 1.118317647580745338925416776501037433975 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{z - t}{y}}{\frac{1}{x}} + t\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z - t}{\frac{y}{x}}\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.664918217275974893528006684776919727564 \cdot 10^{-24}:\\
\;\;\;\;t + \frac{z - t}{\frac{y}{x}}\\

\mathbf{elif}\;t \le -1.336117363059674131707044966713149076742 \cdot 10^{-223}:\\
\;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\

\mathbf{elif}\;t \le 7.81875512884679996695346065746985015342 \cdot 10^{-283}:\\
\;\;\;\;\left(z - t\right) \cdot \frac{x}{y} + t\\

\mathbf{elif}\;t \le 1.118317647580745338925416776501037433975 \cdot 10^{-19}:\\
\;\;\;\;\frac{\frac{z - t}{y}}{\frac{1}{x}} + t\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r25051317 = x;
        double r25051318 = y;
        double r25051319 = r25051317 / r25051318;
        double r25051320 = z;
        double r25051321 = t;
        double r25051322 = r25051320 - r25051321;
        double r25051323 = r25051319 * r25051322;
        double r25051324 = r25051323 + r25051321;
        return r25051324;
}
↓
double f(double x, double y, double z, double t) {
        double r25051325 = t;
        double r25051326 = -2.664918217275975e-24;
        bool r25051327 = r25051325 <= r25051326;
        double r25051328 = z;
        double r25051329 = r25051328 - r25051325;
        double r25051330 = y;
        double r25051331 = x;
        double r25051332 = r25051330 / r25051331;
        double r25051333 = r25051329 / r25051332;
        double r25051334 = r25051325 + r25051333;
        double r25051335 = -1.3361173630596741e-223;
        bool r25051336 = r25051325 <= r25051335;
        double r25051337 = r25051331 * r25051329;
        double r25051338 = r25051337 / r25051330;
        double r25051339 = r25051338 + r25051325;
        double r25051340 = 7.8187551288468e-283;
        bool r25051341 = r25051325 <= r25051340;
        double r25051342 = r25051331 / r25051330;
        double r25051343 = r25051329 * r25051342;
        double r25051344 = r25051343 + r25051325;
        double r25051345 = 1.1183176475807453e-19;
        bool r25051346 = r25051325 <= r25051345;
        double r25051347 = r25051329 / r25051330;
        double r25051348 = 1.0;
        double r25051349 = r25051348 / r25051331;
        double r25051350 = r25051347 / r25051349;
        double r25051351 = r25051350 + r25051325;
        double r25051352 = r25051346 ? r25051351 : r25051334;
        double r25051353 = r25051341 ? r25051344 : r25051352;
        double r25051354 = r25051336 ? r25051339 : r25051353;
        double r25051355 = r25051327 ? r25051334 : r25051354;
        return r25051355;
}

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

Original2.1
Target2.3
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -2.664918217275975e-24 or 1.1183176475807453e-19 < t

    1. Initial program 0.1

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

    2. Taylor expanded around 0 8.5

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

    3. Simplified0.1

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

    if -2.664918217275975e-24 < t < -1.3361173630596741e-223

    1. Initial program 3.2

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

    3. Applied associate-*l/4.5

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

    if -1.3361173630596741e-223 < t < 7.8187551288468e-283

    1. Initial program 6.7

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

    if 7.8187551288468e-283 < t < 1.1183176475807453e-19

    1. Initial program 3.3

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

    2. Taylor expanded around 0 3.6

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

    3. Simplified3.2

      \[\leadsto \color{blue}{\frac{z - t}{\frac{y}{x}}} + t\]
    4. Using strategy rm

    5. Applied div-inv3.3

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

    6. Applied associate-/r*4.3

      \[\leadsto \color{blue}{\frac{\frac{z - t}{y}}{\frac{1}{x}}} + t\]
  3. Recombined 4 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.664918217275974893528006684776919727564 \cdot 10^{-24}:\\ \;\;\;\;t + \frac{z - t}{\frac{y}{x}}\\ \mathbf{elif}\;t \le -1.336117363059674131707044966713149076742 \cdot 10^{-223}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{elif}\;t \le 7.81875512884679996695346065746985015342 \cdot 10^{-283}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{x}{y} + t\\ \mathbf{elif}\;t \le 1.118317647580745338925416776501037433975 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{z - t}{y}}{\frac{1}{x}} + t\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z - t}{\frac{y}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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))