Average Error: 7.0 → 2.2
Time: 48.9s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.921093048146768407671721348483596322734 \cdot 10^{65}:\\ \;\;\;\;\frac{\frac{2}{\frac{z}{x}}}{y - t}\\ \mathbf{elif}\;z \le 1244408611296900.5:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{z}{x}}}{y - t}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.921093048146768407671721348483596322734 \cdot 10^{65}:\\
\;\;\;\;\frac{\frac{2}{\frac{z}{x}}}{y - t}\\

\mathbf{elif}\;z \le 1244408611296900.5:\\
\;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\frac{z}{x}}}{y - t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r26099430 = x;
        double r26099431 = 2.0;
        double r26099432 = r26099430 * r26099431;
        double r26099433 = y;
        double r26099434 = z;
        double r26099435 = r26099433 * r26099434;
        double r26099436 = t;
        double r26099437 = r26099436 * r26099434;
        double r26099438 = r26099435 - r26099437;
        double r26099439 = r26099432 / r26099438;
        return r26099439;
}
↓
double f(double x, double y, double z, double t) {
        double r26099440 = z;
        double r26099441 = -5.921093048146768e+65;
        bool r26099442 = r26099440 <= r26099441;
        double r26099443 = 2.0;
        double r26099444 = x;
        double r26099445 = r26099440 / r26099444;
        double r26099446 = r26099443 / r26099445;
        double r26099447 = y;
        double r26099448 = t;
        double r26099449 = r26099447 - r26099448;
        double r26099450 = r26099446 / r26099449;
        double r26099451 = 1244408611296900.5;
        bool r26099452 = r26099440 <= r26099451;
        double r26099453 = r26099443 * r26099444;
        double r26099454 = r26099449 * r26099440;
        double r26099455 = r26099453 / r26099454;
        double r26099456 = r26099452 ? r26099455 : r26099450;
        double r26099457 = r26099442 ? r26099450 : r26099456;
        return r26099457;
}

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

Original7.0
Target2.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.921093048146768e+65 or 1244408611296900.5 < z

    1. Initial program 12.2

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified2.0

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{z}{x}}}{y - t}}\]

    if -5.921093048146768e+65 < z < 1244408611296900.5

    1. Initial program 2.3

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified9.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{z}{x}}}{y - t}}\]
    3. Using strategy rm

    4. Applied associate-/r/9.3

      \[\leadsto \frac{\color{blue}{\frac{2}{z} \cdot x}}{y - t}\]
    5. Using strategy rm

    6. Applied associate-*l/9.2

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot x}{z}}}{y - t}\]

    7. Applied associate-/l/2.3

      \[\leadsto \color{blue}{\frac{2 \cdot x}{\left(y - t\right) \cdot z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.921093048146768407671721348483596322734 \cdot 10^{65}:\\ \;\;\;\;\frac{\frac{2}{\frac{z}{x}}}{y - t}\\ \mathbf{elif}\;z \le 1244408611296900.5:\\ \;\;\;\;\frac{2 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{z}{x}}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))