Average Error: 6.7 → 0.3
Time: 1.3m
Precision: 64
\[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \le -7.495229651452796 \cdot 10^{+234}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le -3.656744704766576 \cdot 10^{-231}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 2.5855176954021 \cdot 10^{-311}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 3.36468310649941 \cdot 10^{+210}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \end{array}\]
\frac{x \cdot 2.0}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot z - t \cdot z \le -7.495229651452796 \cdot 10^{+234}:\\
\;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\

\mathbf{elif}\;y \cdot z - t \cdot z \le -3.656744704766576 \cdot 10^{-231}:\\
\;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\

\mathbf{elif}\;y \cdot z - t \cdot z \le 2.5855176954021 \cdot 10^{-311}:\\
\;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\

\mathbf{elif}\;y \cdot z - t \cdot z \le 3.36468310649941 \cdot 10^{+210}:\\
\;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r18149323 = x;
        double r18149324 = 2.0;
        double r18149325 = r18149323 * r18149324;
        double r18149326 = y;
        double r18149327 = z;
        double r18149328 = r18149326 * r18149327;
        double r18149329 = t;
        double r18149330 = r18149329 * r18149327;
        double r18149331 = r18149328 - r18149330;
        double r18149332 = r18149325 / r18149331;
        return r18149332;
}
↓
double f(double x, double y, double z, double t) {
        double r18149333 = y;
        double r18149334 = z;
        double r18149335 = r18149333 * r18149334;
        double r18149336 = t;
        double r18149337 = r18149336 * r18149334;
        double r18149338 = r18149335 - r18149337;
        double r18149339 = -7.495229651452796e+234;
        bool r18149340 = r18149338 <= r18149339;
        double r18149341 = 2.0;
        double r18149342 = x;
        double r18149343 = r18149342 / r18149334;
        double r18149344 = r18149341 * r18149343;
        double r18149345 = r18149333 - r18149336;
        double r18149346 = r18149344 / r18149345;
        double r18149347 = -3.656744704766576e-231;
        bool r18149348 = r18149338 <= r18149347;
        double r18149349 = r18149342 * r18149341;
        double r18149350 = r18149349 / r18149338;
        double r18149351 = 2.5855176954021e-311;
        bool r18149352 = r18149338 <= r18149351;
        double r18149353 = 3.36468310649941e+210;
        bool r18149354 = r18149338 <= r18149353;
        double r18149355 = r18149354 ? r18149350 : r18149346;
        double r18149356 = r18149352 ? r18149346 : r18149355;
        double r18149357 = r18149348 ? r18149350 : r18149356;
        double r18149358 = r18149340 ? r18149346 : r18149357;
        return r18149358;
}

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

Original6.7
Target2.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \mathbf{elif}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2.0}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* y z) (* t z)) < -7.495229651452796e+234 or -3.656744704766576e-231 < (- (* y z) (* t z)) < 2.5855176954021e-311 or 3.36468310649941e+210 < (- (* y z) (* t z))

    1. Initial program 17.4

      \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]

    2. Simplified0.5

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\frac{z}{x}}}{y - t}}\]

    3. Taylor expanded around 0 0.4

      \[\leadsto \frac{\color{blue}{2.0 \cdot \frac{x}{z}}}{y - t}\]

    if -7.495229651452796e+234 < (- (* y z) (* t z)) < -3.656744704766576e-231 or 2.5855176954021e-311 < (- (* y z) (* t z)) < 3.36468310649941e+210

    1. Initial program 0.2

      \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \le -7.495229651452796 \cdot 10^{+234}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le -3.656744704766576 \cdot 10^{-231}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 2.5855176954021 \cdot 10^{-311}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 3.36468310649941 \cdot 10^{+210}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))