Average Error: 6.9 → 3.2
Time: 14.6s
Precision: 64
\[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.31181793433353 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2.0}{z}\\ \mathbf{elif}\;x \le 2.042269423331799 \cdot 10^{-231}:\\ \;\;\;\;\frac{2.0 \cdot x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{z}}{\frac{y - t}{x}}\\ \end{array}\]
\frac{x \cdot 2.0}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.31181793433353 \cdot 10^{-58}:\\
\;\;\;\;\frac{x}{y - t} \cdot \frac{2.0}{z}\\

\mathbf{elif}\;x \le 2.042269423331799 \cdot 10^{-231}:\\
\;\;\;\;\frac{2.0 \cdot x}{z \cdot \left(y - t\right)}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r9995259 = x;
        double r9995260 = 2.0;
        double r9995261 = r9995259 * r9995260;
        double r9995262 = y;
        double r9995263 = z;
        double r9995264 = r9995262 * r9995263;
        double r9995265 = t;
        double r9995266 = r9995265 * r9995263;
        double r9995267 = r9995264 - r9995266;
        double r9995268 = r9995261 / r9995267;
        return r9995268;
}
↓
double f(double x, double y, double z, double t) {
        double r9995269 = x;
        double r9995270 = -8.31181793433353e-58;
        bool r9995271 = r9995269 <= r9995270;
        double r9995272 = y;
        double r9995273 = t;
        double r9995274 = r9995272 - r9995273;
        double r9995275 = r9995269 / r9995274;
        double r9995276 = 2.0;
        double r9995277 = z;
        double r9995278 = r9995276 / r9995277;
        double r9995279 = r9995275 * r9995278;
        double r9995280 = 2.042269423331799e-231;
        bool r9995281 = r9995269 <= r9995280;
        double r9995282 = r9995276 * r9995269;
        double r9995283 = r9995277 * r9995274;
        double r9995284 = r9995282 / r9995283;
        double r9995285 = r9995274 / r9995269;
        double r9995286 = r9995278 / r9995285;
        double r9995287 = r9995281 ? r9995284 : r9995286;
        double r9995288 = r9995271 ? r9995279 : r9995287;
        return r9995288;
}

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

Original6.9
Target2.2
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \mathbf{elif}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2.0}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -8.31181793433353e-58

    1. Initial program 9.7

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

    2. Simplified8.4

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\frac{z}{x}}}{y - t}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity8.4

      \[\leadsto \frac{\frac{2.0}{\frac{z}{x}}}{\color{blue}{1 \cdot \left(y - t\right)}}\]

    5. Applied associate-/r/8.3

      \[\leadsto \frac{\color{blue}{\frac{2.0}{z} \cdot x}}{1 \cdot \left(y - t\right)}\]

    6. Applied times-frac2.7

      \[\leadsto \color{blue}{\frac{\frac{2.0}{z}}{1} \cdot \frac{x}{y - t}}\]

    7. Simplified2.7

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

    if -8.31181793433353e-58 < x < 2.042269423331799e-231

    1. Initial program 3.3

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

    2. Simplified2.9

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

    3. Taylor expanded around 0 2.6

      \[\leadsto \frac{\color{blue}{2.0 \cdot \frac{x}{z}}}{y - t}\]
    4. Using strategy rm

    5. Applied associate-*r/2.6

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

    6. Applied associate-/l/1.9

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

    if 2.042269423331799e-231 < x

    1. Initial program 7.6

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

    2. Simplified5.9

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\frac{z}{x}}}{y - t}}\]
    3. Using strategy rm

    4. Applied associate-/r/5.8

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

    5. Applied associate-/l*4.5

      \[\leadsto \color{blue}{\frac{\frac{2.0}{z}}{\frac{y - t}{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.31181793433353 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2.0}{z}\\ \mathbf{elif}\;x \le 2.042269423331799 \cdot 10^{-231}:\\ \;\;\;\;\frac{2.0 \cdot x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{z}}{\frac{y - t}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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))))